Basic Math Examples

{(\frac{7 r - 7 y}{r + y})}^{2} \div (49 r^{2} - 49 y^{2})

${(\frac{7 r - 7 y}{r + y})}^{2} \div (49 r^{2} - 49 y^{2})$

Step 1

Rewrite the division as a fraction.

\frac{{(\frac{7 r - 7 y}{r + y})}^{2}}{49 r^{2} - 49 y^{2}}

$\frac{{(\frac{7 r - 7 y}{r + y})}^{2}}{49 r^{2} - 49 y^{2}}$

Step 2

Factor

7

$7$ out of

7 r - 7 y

$7 r - 7 y$ .

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Step 2.1

Factor

7

$7$ out of

7 r

$7 r$ .

\frac{{(\frac{7 (r) - 7 y}{r + y})}^{2}}{49 r^{2} - 49 y^{2}}

$\frac{{(\frac{7 (r) - 7 y}{r + y})}^{2}}{49 r^{2} - 49 y^{2}}$

Step 2.2

Factor

7

$7$ out of

- 7 y

$- 7 y$ .

\frac{{(\frac{7 (r) + 7 (- y)}{r + y})}^{2}}{49 r^{2} - 49 y^{2}}

$\frac{{(\frac{7 (r) + 7 (- y)}{r + y})}^{2}}{49 r^{2} - 49 y^{2}}$

Step 2.3

Factor

7

$7$ out of

7 (r) + 7 (- y)

$7 (r) + 7 (- y)$ .

\frac{{(\frac{7 (r - y)}{r + y})}^{2}}{49 r^{2} - 49 y^{2}}

$\frac{{(\frac{7 (r - y)}{r + y})}^{2}}{49 r^{2} - 49 y^{2}}$

\frac{{(\frac{7 (r - y)}{r + y})}^{2}}{49 r^{2} - 49 y^{2}}

$\frac{{(\frac{7 (r - y)}{r + y})}^{2}}{49 r^{2} - 49 y^{2}}$

Step 3

Simplify the numerator.

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Step 3.1

Apply the product rule to

\frac{7 (r - y)}{r + y}

$\frac{7 (r - y)}{r + y}$ .

\frac{\frac{{(7 (r - y))}^{2}}{{(r + y)}^{2}}}{49 r^{2} - 49 y^{2}}

$\frac{\frac{{(7 (r - y))}^{2}}{{(r + y)}^{2}}}{49 r^{2} - 49 y^{2}}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Apply the product rule to

7 (r - y)

$7 (r - y)$ .

\frac{\frac{7^{2} {(r - y)}^{2}}{{(r + y)}^{2}}}{49 r^{2} - 49 y^{2}}

$\frac{\frac{7^{2} {(r - y)}^{2}}{{(r + y)}^{2}}}{49 r^{2} - 49 y^{2}}$

Step 3.2.2

Raise

7

$7$ to the power of

2

$2$ .

\frac{\frac{49 {(r - y)}^{2}}{{(r + y)}^{2}}}{49 r^{2} - 49 y^{2}}

$\frac{\frac{49 {(r - y)}^{2}}{{(r + y)}^{2}}}{49 r^{2} - 49 y^{2}}$

\frac{\frac{49 {(r - y)}^{2}}{{(r + y)}^{2}}}{49 r^{2} - 49 y^{2}}

$\frac{\frac{49 {(r - y)}^{2}}{{(r + y)}^{2}}}{49 r^{2} - 49 y^{2}}$

\frac{\frac{49 {(r - y)}^{2}}{{(r + y)}^{2}}}{49 r^{2} - 49 y^{2}}

$\frac{\frac{49 {(r - y)}^{2}}{{(r + y)}^{2}}}{49 r^{2} - 49 y^{2}}$

Step 4

Simplify the denominator.

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Step 4.1

Factor

49

$49$ out of

49 r^{2} - 49 y^{2}

$49 r^{2} - 49 y^{2}$ .

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Step 4.1.1

Factor

49

$49$ out of

49 r^{2}

$49 r^{2}$ .

\frac{\frac{49 {(r - y)}^{2}}{{(r + y)}^{2}}}{49 (r^{2}) - 49 y^{2}}

$\frac{\frac{49 {(r - y)}^{2}}{{(r + y)}^{2}}}{49 (r^{2}) - 49 y^{2}}$

Step 4.1.2

Factor

49

$49$ out of

- 49 y^{2}

$- 49 y^{2}$ .

\frac{\frac{49 {(r - y)}^{2}}{{(r + y)}^{2}}}{49 (r^{2}) + 49 (- y^{2})}

$\frac{\frac{49 {(r - y)}^{2}}{{(r + y)}^{2}}}{49 (r^{2}) + 49 (- y^{2})}$

Step 4.1.3

Factor

49

$49$ out of

49 (r^{2}) + 49 (- y^{2})

$49 (r^{2}) + 49 (- y^{2})$ .

\frac{\frac{49 {(r - y)}^{2}}{{(r + y)}^{2}}}{49 (r^{2} - y^{2})}

$\frac{\frac{49 {(r - y)}^{2}}{{(r + y)}^{2}}}{49 (r^{2} - y^{2})}$

\frac{\frac{49 {(r - y)}^{2}}{{(r + y)}^{2}}}{49 (r^{2} - y^{2})}

$\frac{\frac{49 {(r - y)}^{2}}{{(r + y)}^{2}}}{49 (r^{2} - y^{2})}$

Step 4.2

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = r

$a = r$ and

b = y

$b = y$ .

\frac{\frac{49 {(r - y)}^{2}}{{(r + y)}^{2}}}{49 (r + y) (r - y)}

$\frac{\frac{49 {(r - y)}^{2}}{{(r + y)}^{2}}}{49 (r + y) (r - y)}$

\frac{\frac{49 {(r - y)}^{2}}{{(r + y)}^{2}}}{49 (r + y) (r - y)}

$\frac{\frac{49 {(r - y)}^{2}}{{(r + y)}^{2}}}{49 (r + y) (r - y)}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

\frac{49 {(r - y)}^{2}}{{(r + y)}^{2}} \cdot \frac{1}{49 (r + y) (r - y)}

$\frac{49 {(r - y)}^{2}}{{(r + y)}^{2}} \cdot \frac{1}{49 (r + y) (r - y)}$

Step 6

Combine.

\frac{49 {(r - y)}^{2} \cdot 1}{{(r + y)}^{2} (49 (r + y) (r - y))}

$\frac{49 {(r - y)}^{2} \cdot 1}{{(r + y)}^{2} (49 (r + y) (r - y))}$

Step 7

Multiply

{(r + y)}^{2}

${(r + y)}^{2}$ by

r + y

$r + y$ by adding the exponents.

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Step 7.1

Move

r + y

$r + y$ .

\frac{49 {(r - y)}^{2} \cdot 1}{(r + y) {(r + y)}^{2} (49 (r - y))}

$\frac{49 {(r - y)}^{2} \cdot 1}{(r + y) {(r + y)}^{2} (49 (r - y))}$

Step 7.2

Multiply

r + y

$r + y$ by

{(r + y)}^{2}

${(r + y)}^{2}$ .

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Step 7.2.1

Raise

r + y

$r + y$ to the power of

1

$1$ .

\frac{49 {(r - y)}^{2} \cdot 1}{{(r + y)}^{1} {(r + y)}^{2} (49 (r - y))}

$\frac{49 {(r - y)}^{2} \cdot 1}{{(r + y)}^{1} {(r + y)}^{2} (49 (r - y))}$

Step 7.2.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

\frac{49 {(r - y)}^{2} \cdot 1}{{(r + y)}^{1 + 2} (49 (r - y))}

$\frac{49 {(r - y)}^{2} \cdot 1}{{(r + y)}^{1 + 2} (49 (r - y))}$

\frac{49 {(r - y)}^{2} \cdot 1}{{(r + y)}^{1 + 2} (49 (r - y))}

$\frac{49 {(r - y)}^{2} \cdot 1}{{(r + y)}^{1 + 2} (49 (r - y))}$

Step 7.3

Add

1

$1$ and

2

$2$ .

\frac{49 {(r - y)}^{2} \cdot 1}{{(r + y)}^{3} (49 (r - y))}

$\frac{49 {(r - y)}^{2} \cdot 1}{{(r + y)}^{3} (49 (r - y))}$

\frac{49 {(r - y)}^{2} \cdot 1}{{(r + y)}^{3} (49 (r - y))}

$\frac{49 {(r - y)}^{2} \cdot 1}{{(r + y)}^{3} (49 (r - y))}$

Step 8

Multiply

49 {(r - y)}^{2}

$49 {(r - y)}^{2}$ by

1

$1$ .

\frac{49 {(r - y)}^{2}}{{(r + y)}^{3} (49 (r - y))}

$\frac{49 {(r - y)}^{2}}{{(r + y)}^{3} (49 (r - y))}$

Step 9

Cancel the common factor of

49

$49$ .

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Step 9.1

Cancel the common factor.

$\frac{49 {(r - y)}^{2}}{{(r + y)}^{3} (49 (r - y))}$

Step 9.2

Rewrite the expression.

$\frac{{(r - y)}^{2}}{{(r + y)}^{3} (r - y)}$

Step 10

Cancel the common factor of

${(r - y)}^{2}$ and

$r - y$ .

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Step 10.1

Factor

$r - y$ out of

${(r - y)}^{2}$ .

$\frac{(r - y) (r - y)}{{(r + y)}^{3} (r - y)}$

Step 10.2

Cancel the common factors.

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Step 10.2.1

Factor

$r - y$ out of

${(r + y)}^{3} (r - y)$ .

$\frac{(r - y) (r - y)}{(r - y) {(r + y)}^{3}}$

Step 10.2.2

Cancel the common factor.

$\frac{(r - y) (r - y)}{(r - y) {(r + y)}^{3}}$

Step 10.2.3

Rewrite the expression.

$\frac{r - y}{{(r + y)}^{3}}$